8.13: Fundamental Theorem of Calculus

Velocity due to gravity can be easily calculated by the formula:
v
=
gt
, where
g
is the acceleration due to gravity (9.8m/s
2
) and
t
is time in seconds. In fact, a decent approximation can be calculated in your head easily by rounding 9.8 to 10 so you can just add a decimal place to the time.

Using this function for velocity, how could you find a function that represented the position of the object after a given time? What about a function that represented the instantaneous acceleration of the object at a given time?

Watch This

Guidance

If you think that evaluating areas under curves is a tedious process you are probably right. Fortunately, there is an easier method. In this section, we shall give a general method of evaluating definite integrals (area under the curve) by using antiderivatives.

There are rules for finding the antiderivatives of simple power functions such as
f
(
x
) =
x
2
. As you read through them, try to think about why they make sense, keeping in mind that differentiation reverses integration.

Rules of Finding the Antiderivatives of Power Functions

The Power Rule

where
C
is constant of integration and
n
is a rational number not equal to -1.

A Constant Multiple of a Function Rule

where
k
is a constant.

Sum and Difference Rule

The Constant Rule

where
k
is a constant. (Notice that this rule comes as a result of the power rule above.)

The Fundamental Theorem of Calculus

The fundamental theorem of calculus makes the relationship between derivatives and integrals clear. Integration performed on a function can be reversed by differentiation.

The Fundamental Theorem of Calculus

If a function
f
(
x
) is defined over the interval [
a
,
b
] and if
F
(
x
) is the antidervative of
f
on [
a,
b
], then

We can use the relationship between differentiation and integration outlined in the fundamental theorem of calculus to compute definite integrals more quickly.

Example A

Evaluate

Solution:

This integral tells us to evaluate the area under the curve
f
(
x
) =
x
2
, which is a parabola over the interval [1, 2], as shown in the figure below.

To compute the integral according to the fundamental theorem of calculus, we need to find the antiderivative of
f
(
x
) =
x
2
. It turns out to be
F
(
x
) = (1/3)
x
3
+
C
, where
C
is a constant of integration. How can we get this? Think about the functions that will have derivatives of
x
2
. Take the derivative of
F
(
x
) to check that we have found such a function. (For more specific rules, see the box after this example). Substituting into the fundamental theorem,